Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞

By Evarist Gin'e, Richard Nickl. 2011

113 citations recorded by the source corpus.

The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking $L^r$-norm neighborhoods, $1\le r\le\infty$, of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The rates of contraction are minimax-optimal for $1\le r\le2$, but deteriorate as $r$ increases beyond 2. In the case of Gaussian nonparametric regression a Gaussian prior is devised for which the posterior contracts at the optimal rate in all $L^r$-norms, $1\le r\le\infty$.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ea7a17caa35dcd939931cf425e0b31a798d3ce52

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Citations in the corpus

Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems (2012, methodology): “T] 23 Apr 2013 hand, the area of nonparametric Bayesian statistical estimation and in particular the problem of posterior consistency has attracted a lot of interest in recent years; see for instance [10,24,23,26,27,11,7]. Despite this, the formulation of many of these PDE inverse problems using the”
Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, methodology): “In these situations, to control remainder terms, one may use contraction results in L,2 < p ≤ ∞, instead of L2, such as the ones in [23].”
Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, background): “The Hellinger and total variation norms are by far the most common metrics for establishing posterior contraction rates and there is only a recent small literature for stronger norms [9, 22, 23, 53].”
Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions (2015, methodology): “By Richard Nickl and Jakob Söhl”
On adaptive posterior concentration rates (2013, background): “n−(β−1/2)/(2β+1), this explains the (suboptimal) rate observed in [17] which was derived using the standard approach, and thus a bound of the type (5.”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, background): “Castillo and Nickl [3] showed that for the Gaussian white noise model a BvM theorem can hold in weaker topologies for some natural priors, and the resulting credible sets appropriately modified will have asymptotically the correct coverage and optimal size.”
Adaptive confidence sets in $$L^2$$ (2011, unknown): “To simplify the exposition we prove the result for ‖ f ‖∞ known, otherwise the result follows from the same proof, with ‖ f ‖∞ replaced by ‖ fn( jmax)‖∞, a consistent estimator for ‖ f ‖∞ that satisfies sufficiently tight uniform exponential error bounds (using inequality (26) in [15] and proceeding”